Hypergeometric function identities Hypergeometric function lists identities for the Gaussian hypergeometric function; Generalized hypergeometric function lists identities for more general hypergeometric functions; Bailey's list is a list of the hypergeometric function identities in Bailey (1935) given by Koepf (1995). Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials. 7) is given and if a0 6= 0, then it is easy to Mar 5, 2025 · A relation expressing a sum potentially involving binomial coefficients, factorials, rational functions, and power functions in terms of a simple result. There are many approaches to these functions and the literature can fill books. 2 Definition and Analytic Properties; 16. In mathematics, hypergeometric identities are equalities involving sums over hypergeometric terms, i. The main references used in writing this chapter are Andrews et al. It allows us to prove many generalizations of recent results, and also to evaluate the sum of several families of Ramanujan-type series for $${1}/{\\pi }$$ 1 / π , $$\\sqrt{2}/\\pi $$ 2 / π and $$\\sqrt{3}/\\pi $$ 3 / π . 在数学中，高斯超几何函数或普通超几何函数2F1(a,b;c;z)是一个用超几何级数定义的函数，很多特殊函数都是它的特例或极限。所有具有三个正则奇点的二阶线性常微分方程的解都可以用超几何函数表示。 Feb 19, 2021 · In this chapter we focus on the hypergeometric function 2 F 1 or Gauss hypergeometric function for complex parameters and arguments. 6 Transformations of Variable; 16. 23 This book presents a geometric theory of complex analytic integrals representing hypergeometric functions of several variables. DRIVER AND S. 1). In 1812 Gauss published a study of the hypergeometric series Sep 3, 2024 · However, we will give formulas for this extension in the cases of Gauss hypergeometric function (Chap. 3. has the series expansion , Feb 19, 2021 · In this chapter we focus on the hypergeometric function 2 F 1 or Gauss hypergeometric function for complex parameters and arguments. J. E. This method is based upon our recent convolution theorem and some classical hypergeometric identities. Finally (1. Starting from an integrand which is a product of powers of polynomials, integrals are explained, in an open affine space, as a pair of twisted de Rham cohomology and its dual over the coefficients of local system. 2), is recalled to be the Pochhammer symbol indi-cating a shifted factorial. This is also known as the confluent hypergeometric function of the first kind. The main goal of this paper is to derive a number of identities for the generalized hypergeometric function evaluated at unity and for certain terminating multivariate hyper-geometric functions from the symmetries and other properties of Meijer’s Gfunction. A generalized hypergeometric function _pF_q(a_1,,a_p;b_1,,b_q;x) is a function which can be defined in the form of a hypergeometric series, i. Hundreds of thousands of mathematical results derived at Wolfram Research give the Wolfram Language unprecedented strength in the transformation and simplification of hypergeometric functions. The author thanks Richard Askey and Simon Ruijsenaars for many helpful recommendations. The con uent hypergeometric functions 0F1(z) and 1F1(z) form exponential integrals, incomplete gamma functions, Bessel functions, and related functions. These functions generalize the classical hypergeometric functions of Gauss, Horn, Appell, and Lauricella. Case p q: the series converges for any zand de nes an entire function with an irregular (exponential) singularity at z= 1. The Whittaker Functions give an alternative form of the solution. In mathematics, the Gaussian or ordinary hypergeometric function 2 F 1 (a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. For examples see §§ 14. Many of them can be applied to the nite eld settings to obtain very analogous formulas. Identities (151 formulas) Hypergeometric2F1. Complete definition 07. \nonumber \] The purpose of this section is only to introduce the hypergeometric function. Thanks to results by Fasenmyer, Gosper, Zeilberger, Wilf, and Petkovšek, the problem of determining whether a given hypergeometric sum is expressible in simple closed form and, if so, finding the form, is now (subject to a mild restriction This textbook provides an elementary introduction to hypergeometric functions, which generalize the usual elementary functions. In nine comprehensive chapters, Dr. The Kampé de Fériet function can represent derivatives of generalized hypergeometric functions with respect to their parameters, as well as indefinite integrals of two Assuming "hypergeometric functions" is referring to a mathematical definition identities 1F1(a,c,z) Whipple's transformation; identities 2F1(a,b,c,z) Feb 26, 2025 · We reinvestigate the Calogero-Sutherland-type (CS-type) models and generalized hypergeometric functions. Note that the two definitions coincide when , including the common case . To analyze the properties of CS operators and the generalized hypergeometric functions, the $\\hat W$-operators and $\\hat O generalized hypergeometric functions (8) will be used to prove identities for these functions. To derive the hypergeometric function based on the Hypergeometric Differential Equation, plug In mathematics, the Gaussian or ordinary hypergeometric function 2 F 1 (a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. We construct the generalized CS operators for circular, Hermite, Laguerre, Jacobi and Bessel cases and establish the generalized Lasselle-Nekrasov correspondence. HermiteH[nu,z] (229 formulas) ParabolicCylinderD[nu,z] (235 formulas) Finally, the usefulness of both ordinary and basic hypergeometric functions in the proof of and classification of combinatorial identities, and some of the recent applications of basic hypergeometric functions in physics are discussed. Consecutive neighbors (nine basic relations) Distant This chapter is based in part on Chapter 15 of Abramowitz and Stegun (1964) by Fritz Oberhettinger. 1 Diﬀerential equations and systems of equa-tions A diﬀerential ﬁeld K is a ﬁeld equipped with a derivation, that is, a map ∂: Apr 18, 2013 · We describe a method of obtaining weighted norm inequalities for generalized hypergeometric functions. Many of the nonelementary functions that arise in mathematics and physics also have representations as hypergeometric series. 117). We then illustrate how the confluent hypergeometric functions induce Bessel functions, as well as many other useful functions in mathematical physics and probability theories. Continued fraction representations (2 formulas) Differential equations (9 formulas) Transformations (3 formulas) Identities (24 formulas) Differentiation (22 formulas) Integration (3 formulas) Summation (2 formulas) Operations (7 formulas) Representations through more general functions (6 formulas) Theorems (0 formulas) History (0 formulas) hypergeometric function and their transformations. Many other special functions are related to the hypergeometric function after making some variable transformations. Also, several families of series involving the central binomial Hypergeometric Functions: Hypergeometric2F1Regularized[a,b,c,z] (865 formulas)Primary definition (8 formulas) Specific values (222 formulas) Any hypergeometric function for which a quadratic transformation exists can be expressed in terms of associated Legendre functions or Ferrers functions. 4 Identities with generalized hypergeometric functions One of the main results of this paper are the following propositions for the generalized hypergeometric functions (8). This is the most common form and is often called the hypergeometric function. Maybe the answer lies in the determinant form of this identity. Askey, and R. 5) where (·) k, as deﬁned by Eq. 5 Integral Representations and Integrals; 16. 16. Hypergeometric Functions Hypergeometric2F1[a,b,c,z] 概要. These identities were traditionally found 'by hand'. These identities occur frequently in solutions to combinatorial problems, and also in the analysis of algorithms. Oct 31, 2024 · These formulas are analyzed in light of three Gauss relations for contiguous functions, with the aid of a relation between the Gauss hypergeometric functions and the Lerch transcendent. A number of generalized hypergeometric functions has special names. Hypergeometric formulas over nite elds We mentioned a few techniques for obtaining hypergeometric formulas such as from de nition/setup, from comparing coe cients, from specializing values. cumulative distribution function in terms of hypergeometric functions, it may be possible to obtain new insights or make computations more efficient. MODULAR IDENTITIES AND HYPERGEOMETRIC FUNCTIONS 5 For the hypergeometric functions (1. 6 Recurrence Relations We know by Theorem 3. A series x n is called hypergeometric if the ratio of successive terms x n+1 /x n is a rational function of n. If a hypergeometric series (2. Details. Now we consider a function deﬁned by means of a series given by F(a,b;c;x) = ∞ k=0 (a) k(b) k (c) kk! xk = 1+ ab c ·x + a(a +1)b(b +1) 2!c(c +1) ·x2 +···, (7. Unlike the linear (fractional) transformations of these functions, of which Pfaff's formula in Theorem 2. There exist now them. The transformation formulas between two hypergeometric functions in Group 2, or two hypergeometric functions in Group 3, are the linear transformations (15. 10). is the generalized hypergeometric function . The Euler integral representation of the Gauss hypergeometric function is well known and plays a prominent role in the derivation of transformation identities and in the evaluation of Dec 27, 2019 · Hypergeometric functions, Hypergeometric functions, Hypergeometrische Reihe, Hypergeometrische functies, Fonctions hypergéométriques, Hypergeometrische Reihe, Hypergeometric functions Publisher Cambridge [England] : University Press Collection internetarchivebooks; inlibrary; printdisabled Contributor Internet Archive Language English Item Size General functions of this nature have in fact been developed and are collectively referred to as functions of the hypergeometric type. Recurrence identities. The structure of this article is as follows: - Section 2: A number of random-looking but relevant examples; - Sections 3, 4, 5: Gauss hypergeometric functions and examples of their general-izations; - Section 6: A-hypergeometric functions, a May 24, 2024 · This function satisfies the differential equation \[x y^{\prime \prime}+(\gamma-x) y^{\prime}-\alpha y=0 . See for example Special Functions by G. 21(iii) . In this chapter, we introduce three important approaches to hypergeometric functions. 4 Other versions of several of the identities in this subsection can be constructed with the aid of the operator identity. Hypergeometric Functions HypergeometricPFQ[{a 1,a 2,a 3},{b 1,b 2},z] Identities. 7 Relations to Other Functions; 16. For instance, we recover two- and three-term Thomae relations for 3F Mar 1, 2024 · In this section, we first roughly manipulate the identity (28) to find that it implies a duplication-type summation formula for Kummer’s confluent hypergeometric function of the first kind for a limited parameter region. There are several varieties of these functions, but the most common are the standard hypergeometric function ( which we discuss in this chapter) and the confluent hypergeometric function (Chap. (7. In this paper we show and prove the fundamental (Abramowitz and Stegun 1972, p. Initially this document started as an informal introduction to Gauss’ Mar 5, 2020 · Chu-Vandermonde Identity, Dougall's Formula, Generalized Hypergeometric Function, Hypergeometric Function, Thomae's Theorem Explore with Wolfram|Alpha More things to try: Hypergeometric Functions: HypergeometricPFQ[{a 1,a 2},{b 1,b 2},z] (31337 formulas)Primary definition (2 formulas) Specific values (31206 formulas) Apr 26, 2021 · In the present paper, the authors mainly study the extensions of transformation identities – to the zero-balanced hypergeometric function $F(a,b;a+b;r)$ by showing the monotonicity properties of certain combinations in terms of zero-balanced hypergeometric functions and elementary functions, thus giving complete solutions to the problem of Apr 26, 2021 · In the present paper, the authors mainly study the extensions of transformation identities – to the zero-balanced hypergeometric function $F(a,b;a+b;r)$ by showing the monotonicity properties of certain combinations in terms of zero-balanced hypergeometric functions and elementary functions, thus giving complete solutions to the problem of He started his research in geometric function theory, switching towards orthogonal polynomials and special functions and towards computer algebra. . Question. (For some additional remarks see also Sect. Gauss’ hypergeometric function Frits Beukers October 10, 2009 Abstract We give a basic introduction to the properties of Gauss’ hypergeometric functions, with an emphasis on the determination of the monodromy group of the Gaussian hyperegeo-metric equation. This allows hypergeometric functions for the first time to take their place as a practical nexus between many special functions — and makes possible a Euler–Gauss hypergeometric function: the hypergeometric function F(λ,k;t) associated with a root system R. These values are s = 61 , 14 , 31 . Distant neighbors with respect to q. Hypergeometric Functions: HypergeometricPFQ[{a 1,a 2,a 3},{b 1,b 2},z] (40964 formulas)Primary definition (2 formulas) Specific values (40747 formulas) May 25, 1999 · To confuse matters even more, the term ``hypergeometric function'' is less commonly used to mean Closed Form. 8. The generalized Gauss function is also used in mathematical statistics and the basic analogues of the Gauss functions have applications in the field of number theory. 20) follows after letting m → ∞. Consecutive neighbors. 3(i) – 14. 10 Expansions in Series of F q p Functions May 5, 2013 · Gauss found and analyzed a quadratic transformation of hypergeometric functions; this apparently led him to the problem of monodromy. In particular, it is shown that some product identities involving the divergent hypergeometric series lead to the convergent hypergeometric inequalities. We do this by looking at hypergeometric functions that are at the same time algebraic. WZ pairs are named after Herbert S. 5 is an example, quadratic transformations exist only under certain conditions on the parameters. The trivial example is 0F0(z) = exp(z). 5)istermedasa Hypergeometric Functions (218,254 formulas) Hermite, Parabolic Cylinder, and Laguerre Functions. Andrews, R. May 24, 2024 · Most functions that you know can be expressed using hypergeometric functions. We then move on to a rigorous proof of the identity for more generic arguments. A lot of e orts have been made to deepen understanding of the hypergeometric functions, which have accumulated hundreds of Hypergeometric Functions HypergeometricPFQ[{a 1,a 2},{b 1,b 2},z] Identities (21 formulas) Recurrence identities (4 formulas) Functional identities (17 formulas) There are several common standard forms of confluent hypergeometric functions: Kummer's (confluent hypergeometric) function M(a, b, z), introduced by Kummer , is a solution to Kummer's differential equation. e. In mathematics, specifically combinatorics, a Wilf–Zeilberger pair, or WZ pair, is a pair of functions that can be used to certify certain combinatorial identities. 9 Zeros; 16. On the other hand, the series 1F1(x) converges for all x ∈ and is called Kummer’s conﬂuent hypergeometric function. A function deﬁned by means of Eq. Proposition 2. 505). 8). The relationship between the special functions and their corresponding hypergeometric functions is given in Generalized Hypergeometric Functions. The following identities with the generalized hypergeometric Dec 11, 2023 · The aim of this work is to present an interesting identity relating generalized hypergeometric functions. A lot of e orts have been made to deepen understanding of the hypergeometric functions, which have accumulated hundreds of A generalized hypergeometric function therefore has parameters of type 1 and parameters of type 2. It is a solution of a second-order linear ordinary differential equation (ODE). 4 Argument Unity; 16. Roy, 1999, Cambridge University Press. 6). 8 Differential Equations; 16. These functions generalize the Euler– Gauss hypergeometric function (for the rank one root system) and the ele-mentary spherical functions on a real semisimple Lie group (for particular parameter values). 3 Derivatives and Contiguous Functions; 16. Wilf and Doron Zeilberger , and are instrumental in the evaluation of many sums involving binomial coefficients , factorials , and in general Mar 5, 2025 · The Kampé de Fériet function is a special function that generalizes the generalized hypergeometric function to two variables and includes the Appell hypergeometric function F_1(alpha;beta,beta^';gamma;x,y) as a special case. Four such generalizations were investigated by Lauricella (1893), and more fully by Appell and Kampé de Fériet (1926, p. Rao and Dr. I) Introduction Kummer-type transformations, Racah polynomials, Whipple’s transformation, Wilf–Zeilberger algorithm, applied to generalized hypergeometric functions, contiguous balanced series, contiguous relations, extensions of Kummer’s relations, generalized hypergeometric functions, identities, recurrence relations, relation to generalized LAPLACE APPROXIMATIONS FOR HYPERGEOMETRIC FUNCTIONS WITH MATRIX ARGUMENT1 BY RONALD W. This note is based on Fang-Ting Tu’s course on \Hypergeometric Functions" given at LSU in Fall 2020 and Ling Long’s mini-lectures on \Hypergeometric Functions, Character Sums and Applications" given at University of The hypergeometric functions that correspond to Groups 3 and 4 have a nonlinear function of z as variable. Mar 5, 2025 · Lauricella functions are generalizations of the Gauss hypergeometric functions to multiple variables. Jan 21, 2022 · The asymptotic behaviour of hypergeometric functions for large values of $ | z | $ is completely described by formulas yielding analytic continuations in a neighbourhood of the point $ z = \infty $ , , . Specific values (189 formulas) General characteristics (23 formulas) Series representations (31 formulas) Integral representations (5 formulas) Limit representations (2 formulas) Continued fraction representations (2 formulas) Differential equations (17 formulas) Transformations (3 formulas) Identities (31 formulas) Differentiation (37 formulas) Sep 17, 2024 · The hypergeometric function $_2F_1$ can be rewritten using Euler and Pfaff transformations in the following way: \begin{equation} (1-z)^k \quad _2F_1\left(-k,1,\frac{k+3}{2},\frac{z}{z-1}\right) =\quad _2F_1\left(-k,\frac{k+1}{2},\frac{k+3}{2},z\right) \end{equation} What I would like to know is whether there is a similar relationship to the previous one to simplify the following expression This detailed monograph outlines the fundamental relationships between the hypergeometric function and special functions. Hypergeometric Functions Hypergeometric2F1[a,b,c,z] Identities. 3(iii) and 14. , in quantum dynamics . A number of the new weighted norm Here (a) N is the Pochhammer symbol: (a) 0 = 1, (a) N = a ⋯ (a + N − 1). BUTLER AND ANDREW T. Hypergeometric Functions: Hypergeometric2F1[a,b,c,z] (111951 formulas)Primary definition (8 formulas) Specific values (111271 formulas) Chapter 1 Ordinary linear diﬀerential equations 1. The main technicality lies in analyzing the delta terms. A. 34) 1 1 Fs (x) := 2 F1 ( − s, + s; 1; x) 2 2 there are three other values of s for which similar results hold. The motivation for computing hypergeometric functions will be discussed, with details given of some of the practical applications of these functions Hypergeometric Functions: HypergeometricPFQ[{a 1},{b 1,b 2},z] (21885 formulas)Primary definition (3 formulas) Specific values (21767 formulas) AN INTEGRAL REPRESENTATION OF SOME HYPERGEOMETRIC FUNCTIONS K. Outside of the unit circle €z⁄<1the function 2F1Ha,b;c;zL is defined as the analytic continuation with respect to z of this sum, with the parameters a, b, c held fixed. See also Bateman Function, Confluent Hypergeometric Function of the First Kind, Confluent Hypergeometric Limit Function, Coulomb Wave Function, Cunningham Function, Gordon Function, Hypergeometric Function, Poisson-Charlier Polynomial, Toronto Function Dec 2, 2024 · This book presents a novel journey of the Gauss hypergeometric function and contains the different versions of the Gaussian hypergeometric function, including its classical version. If $ \alpha $, $ \beta $ and $ z $ are given and $ | \gamma | $ is sufficiently large, $ | \mathop{\rm arg} \gamma | < ( \pi - \epsilon The Hypergeometric, Supertrigonometric, and Superhyperbolic Functions In this chapter, we introduce the Euler gamma function, the Pochhammer symbols, the Gaussian hypergeometric series, the Clausen hypergeometric series and the Super-trigonometric, and Superhyperbolic functions via Gaussian hypergeometric series May 5, 2013 · A series Σ c n is hypergeometric if the ratio c n +1 / c n is a rational function of n. AB - The application of basic hypergeometric functions to partitions is briefly discussed. Lakshminarayanan present a unified approach to the study of special functions of mathematics using Group theory. 2 Hypergeometric functions A function f(z) = P 1 k=0 c(k)zk is called hypergeometric if the Taylor coe cients c(k) form a hypergeometric sequence, meaning that they satisfy a rst-order recurrence relation c(k+ 1) = R(k)c(k) where the term ratio R(k) is a rational function of k. (1999) and Temme (1996b). How do I prove ($*$) ‘directly,’ perhaps by using contiguous hypergeometric identities? The gamma function also satis es multiplication formulas. Mathematical function, suitable for both symbolic and numerical manipulation. Note that the function 2F1(x) (whose radius of convergence is 1) was introduced by Gauss and is therefore called Gauss’ hypergeometric function. Oct 5, 2021 · We then introduce the concept of confluent hypergeometric functions, which can be considered as a limiting case of the Gauss hypergeometric function. If p = q = 1 then the function is called a conﬂuent hypergeometric function. May 11, 2021 · The main goal of this paper is to derive a number of identities for generalized hypergeometric function evaluated at unity and for certain terminating multivariate hypergeometric functions from HypergeometricPFQ. Abstract. The hypergeometric functions are solutions to the Hypergeometric Differential Equation, which has a Regular Singular Point at the Origin. Many for-mulae for the function have been established and they are contained in the books on special functions such as [WW], [EMO], [WG], [SW] etc. In this course we will study multivariate hypergeometric functions in the sense of Gel’fand, Kapranov, and Zelevinsky (GKZ systems). 2. A hypergeometric function is called Gaussian if p = 2 and q = 1. g. The intimate connection between hypergeometric functions and the special functions of mathematics has been stated succinctly as a theorem by W W Bell (1968). First, Euler's fractional integral Mar 5, 2025 · Regularized hypergeometric functions are implemented in the Wolfram Language as the functions Hypergeometric0F1Regularized[b, z], Hypergeometric1F1Regularized[a, b, z the family of hypergeometric functions, the most well-known ones are Gauss hypergeometric function and Kummer con uent hypergeometric function, and their natural extension is known as generalized hypergeometric function [1,2]. Mar 7, 2025 · The special functions are extremely useful tools for obtaining closed form as well as series solutions to a variety of problems arising in science and engineering we tryed to reobtain the known results by the new method. 11 that the derivative of ${ }_{r}F_{s}\left ( x\right ) $ with respect to x is (up to a multiplicative constant) an hypergeometric function. Hypergeometric Functions HypergeometricPFQ[{a 1,,a p},{b 1,,b q},z]: Identities (24 formulas) Recurrence identities (1 formula) 2 Background on hypergeometric functions In this section, we will introduce properties of the generalized hypergeometric function that will be exploited in this project. For small , the function behaves as . is called a confluent hypergeometric limit function, and is implemented in the Wolfram Language as Hypergeometric0F1[b, z]. WOOD Colorado State University and University of Nottingham In this paper we present Laplace approximations for two functions of matrix argument: the Type I conﬂuent hypergeometric function and the Gauss hypergeometric function. In the 1990s he has written several books about the use of computer algebra in math education, followed by the first edition of his monograph Hypergeometric Summation. Dr Slater's treatment leads on from a discussion of the Gauss functions to the basic hypergeometric functions, the hypergeometric integrals, bilateral series and Appel series. the family of hypergeometric functions, the most well-known ones are Gauss hypergeometric function and Kummer con uent hypergeometric function, and their natural extension is known as generalized hypergeometric function [1,2]. Specific values (24 formulas) General characteristics (17 formulas) Series representations (20 formulas) Integral representations (5 formulas) Limit representations (3 formulas) Continued fraction representations (2 formulas) Differential equations (6 formulas) Transformations (3 formulas) Identities (22 formulas) Differentiation (31 formulas) The Gauss hypergeometric function is the most fundamental and important spe-cial function and it has long been studied from various points of view. Assuming "hypergeometric function" is referring to a mathematical definition identities 1F1(a,c,z) Have a question about using Wolfram|Alpha? May 31, 2021 · There are now three famous identities, the ‘Dougall-Ramanujan identity’…and the ‘Rogers-Ramanujan identities’, in which he had been anticipated by British mathematicians…As regards hypergeometric series one may say, roughly, that he discovered the formal theory, set out in Bailey’s tract, as it was known up to 1920. Below is a list of hypergeometric identities. Key words and phrases: Hypergeometric functions; distribution theory; chi-square Distribution, Non-centrality Parameter. In addition to their importance as special functions, these generalized hypergeometric functions and their immediate multivariate generalizations are suspected to be able to describe all linear differential equations with “arithmetic” properties—having solutions that are G–functions in Siegel’s sense (that is Hypergeometric Functions Reading Problems Introduction The hypergeometric function F(a, b; c; x) is deﬁned as F(a; b; c; x)= 2F 1(a, b; c; x)=F(b, a; c; x) =1+ ab c x + a(a +1)b(b +1) c(c +1) x2 2! In mathematics, the Gaussian or ordinary hypergeometric function 2 F 1 (a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. Hypergeometric function，超幾何函數。許多 infinite series 都能以這樣的形式表現，於是可以利用 hypergeometric function 的眾多性質，系統性的操作這些 series（包括在 Binomial Coefficient 見到的 series）；以下舉些例子： Hypergeometric Functions Fang-Ting Tu, joint with Alyson Deines, Jenny Fuselier, Ling Long, Holly Swisher a Women in Numbers 3 project National Center for Theoretical Sciences, Taiwan Mini-workshop at LSU on Algebraic Varieties, Hypergeometric series, and Modular Forms Fang Ting Tu (NCTS) Hypergeometric Functions April 14th, 2015 1 / 32 In mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. It includes plenty of solved exercises and it is appropriate for a wide audience, starting from undergraduate students in mathematics, physics and engineering. Many functions have representations as hypergeometric function ; applications can be found, e. For example, a basic set of such relations for Appell's F 1 is given by: For €z⁄<1 and generic parameters a, b, c, the hypergeometric function 2F1Ha,b;c;zL is defined by the above infinite sum (that is convergent). Otherwise the function is called a generalized hypergeometric function. Oct 3, 2024 · But I've been unable to get any contiguous identities to work for this problem. Mar 5, 2025 · This is the -hypergeometric function as defined by Bailey (1935), Slater (1966), Andrews (1986), and Hardy (1999). , a series for which the ratio of successive terms can be written (c_(k+1))/(c_k)=(P(k))/(Q(k))=((k+a_1)(k+a_2)(k+a_p))/((k+b_1)(k+b_2)(k+b_q)(k+1))x. 4) and polylogarithm function (Chap. the coefficients occurring in hypergeometric series. Hypergeometric Functions HypergeometricPFQ[{a 1,a 2,a 3,a 4},{b 1,b 2,b 3},z] Identities (31 formulas) Recurrence identities (5 formulas) Functional identities (26 Like the Gauss hypergeometric series 2 F 1, the Appell double series entail recurrence relations among contiguous functions. JOHNSTON Dedicated to Ed Saff on the occasion of his 60th birthday Abstract. 15 Hypergeometric Function Properties 15. qtpe qjhxan dtlu imbjssp cch rks rnsv dchts nuaweso vlhns wmfvdm mbo noknq tapj wmvo

Hypergeometric function identities. (Abramowitz and Stegun 1972, p.